E. D. Belokolos, F. Gesztesy, K. A. Makarov and L. A. Sakhnovich
Matrix-Valued Generalizations of the Theorems of Borg and Hochstadt
(101K, LaTeX)
ABSTRACT. We prove a generalization of the well-known theorems by Borg and
Hochstadt for periodic self-adjoint Schr\"odinger operators
without a spectral gap, respectively, one gap in their spectrum, in the
matrix-valued context. Our extension of the theorems of Borg and
Hochstadt replaces the periodicity condition of the potential by the more
general property of being reflectionless (the resulting potentials then
automatically turn out to be periodic and we recover Despr\'es' matrix
version of Borg's result). In addition, we assume the spectra to have
uniform maximum multiplicity (a condition automatically fulfilled in the
scalar context considered by Borg and Hochstadt). Moreover, the
connection with the stationary matrix KdV hierarchy is established.
The methods employed in this paper rely on matrix-valued Herglotz
functions, Weyl--Titchmarsh theory, pencils of matrices, and basic inverse spectral theory for matrix-valued Schr\"odinger operators.